The most thorough way to do what you need is to erase the Mac and reinstall its originally installed version of OS X, but assuming you and your spouse are fairly close to one another and merely inconvenienced by getting each other's reminders and so forth (vs. being complete strangers and concerned about the loss of personal information) simply "sign out" of all iCloud services, and sign in again using your own Apple ID.
You may still encounter occasions in which you will need his Apple ID, such as updating software or playing music that he may have purchased using his Apple ID instead of yours.
Go to System Preferences > iCloud, and sign out. Select "Delete from this Mac" when it appears.
Then, sign out of each app that is tied to an Apple ID. For example, in Calendar, go to Preferences > Accounts. Select the account you do not wish to use and then click the [—] (minus) button. If you get an alert that the account is "shared using iCloud", return to System Preferences and de-selct that account.
Then, sign in to iCloud using your own Apple ID.
The brute force method in which you are parting with the Mac to someone outside the family (erasing everything) is probably overkill in your circumstance.